Updated vignettes with new TFR-rate estimations.

R/reproduction_indices.R

@@ -183,7 +183,7 @@ lt.reproduction.mortaar_life_table <- function(life_table, fertility_rate = "BA_
       BA_log = { fertil_rate <- log(indx$p5_19[[1]]) * 1.58341 + 9.45601 },
       { stop(paste(
           "Please choose a valid fertility rate",
-          "(either 'McO', 'BA_linear', 'BA_log' or 'BA_power' or a number)."
+          "(either 'McO', 'TOMc', 'BA_linear', 'BA_log' or 'BA_power' or a number)."
         )) }
     )
   } else if (is.numeric(fertility_rate)) {

vignettes/mortAAR_vignette_extended.Rmd

@@ -12,7 +12,7 @@ vignette: >
 
 ## Introduction
 
-Life tables have only a marginal position in archaeological research, for different reasons. Columns with rows of multidigit numbers carrying strange labels can be daunting, and indeed the mathematics behind the construction of life tables based on modern census data are not for the faint-of-heart. Compared to life tables based on real census data, archaeological ones are more straightforward, but their interpretation carries its own problems. First and foremost, they pretend to have a degree of exactness which is not supported by the underlying data. This is another reason, why life-tables are considered with a certain degree of suspiciousness by archaeologists. And finally, despite the fact that the construction of archaeological life tables can be straightforward, it involves a lot of computation and number-crunching compared to other methods used in the archaeological discourse. To our current knowledge, an easy to use and accessible tool for computing life tables is missing. This is why we sat down and created mortAAR for R. We hope it will be of use for archaeologists world-wide.  
+Life tables have only a marginal position in archaeological research, for different reasons. Columns with rows of multidigit numbers carrying strange labels can be daunting, and indeed the mathematics behind the construction of life tables based on modern census data are not for the faint-of-heart. Compared to life tables based on real census data, archaeological ones are more straightforward, but their interpretation carries its own problems. First and foremost, they pretend to have a degree of exactness which is not supported by the underlying data. This is another reason why life-tables are considered with a certain degree of suspiciousness by archaeologists. And finally, despite the fact that the construction of archaeological life tables can be straightforward, it involves a lot of computation and number-crunching compared to other methods used in the archaeological discourse. To our current knowledge, an easy to use and accessible tool for computing life tables is missing. This is why we sat down and created mortAAR for R. We hope it will be of use for archaeologists world-wide.  
 
 From an outer appearance, life tables from modern census data and archaeological ones, like generated by mortAAR, look very similar. For modern census data, though, there are far more sophisticated measures available. We cannot go into more detail here but refer the reader to the abundant literature on demography and life table construction.
 

vignettes/mortAAR_vignette_lt_correction.Rmd

@@ -50,14 +50,16 @@ In contrast, Bocquet-Appel and Masset took the raw number of dead (D<sub>x</sub>
 
 If either of these prerequisites is not met, the results from such data should be treated with extreme caution as the mortality structure is different from that of known populations. Due to the specific nature of the indices, they only give meaningful results if 5-year-age categories have been chosen for the non-adults.
 
+Quite recently, B. Taylor and M. Oxenham [-@taylor_oxenham_2024] added a comparison of Total fertility rates (TRF) according to different formulas which depend either on subadults or adults. The formualas Taylor and Oxenham used either weigh all individuals aged 0--14 against all individuals or all individuals aged 15--49 against all individuals aged 15+. The formulas differ from the original ones published by C. McFadden and M. Oxenham [-@mcfadden_oxenham_2018a] and B. Taylor et al. [-@taylor_et_al_2023] because the data basis is slighty different. If the results of the formulas deviate by more than 0.692 (the standard error of estimate, SEE), they conclude that there is a problem with the age structure. In contrast to the tests by Weiss and Masset and Bocquet-Appel, there is no need to divide the sample into 5-year-steps, as long as the limits at 15 and 50 years are respected.
+
 As example, we use the data-set of the medieval cemetery of Schleswig.
 
 ```{r}
 schleswig <- life.table(schleswig_ma[c("a", "Dx")])
 lt.representativity(schleswig)
 ```
 
-The result is a dataframe where the individual conditions with the actual results are listed. The last column provides the verdict if the respective condition is met or not. In the case of the Schleswig cemetery, only the Weiss criterium (1) and the Bocquet-Appel/Masset criterium (2) is `TRUE`, the other two are `FALSE`. Therefore, it can be argued that the Schleswig data is not representative of a complete once-living population. However, it might be that the Bocquet-Appel/Masset criterium (2) is too conservative as only two individuals would have to have died at this age to bring the ratio above 2 [@herrmann_prahistorische_1990, 307]. P. Sellier [-@sellier_1989, 23] has even argued that the ratio should be 1.5 instead of 2. In this case, both criteria of Bocquet-Appel and Masset would be met for the Schleswig cemetery.
+The result is a dataframe where the individual conditions with the actual results are listed. The last column provides the verdict if the respective condition is met or not. In the case of the Schleswig cemetery, only the Weiss criterium (1) and the Bocquet-Appel/Masset criterium (2) is `TRUE`, the other three are `FALSE`. Therefore, it can be argued that the Schleswig data is not representative of a complete once-living population. However, it might be that the Bocquet-Appel/Masset criterium (1) is too conservative as only two individuals would have to have died at this age to bring the ratio above 2 [@herrmann_prahistorische_1990, 307]. P. Sellier [-@sellier_1989, 23] has even argued that the ratio should be 1.5 instead of 2. In this case, both criteria of Bocquet-Appel and Masset would be met for the Schleswig cemetery.
 
 ## Correction of life table data after Bocquet-Appel and Masset
 
@@ -91,4 +93,10 @@ life.table(schleswig_ma[c("a", "Dx")])
 
 With the uncorrected data, the youngest age group (years 0--4) 'only' comprises around 20% of the population. However, according to the formulas by Masset and Bocquet-Appel, this number should be more than doubled (46%). This would mean that in reality nearly 50% of the individuals died before they reached their 5^th^ birthday. Applying this value to the life table, the number of individuals increases from 247 to 368 and at the same time the life expectancy at birth decreases from 30.7 to 21.1 years. Please note that this value differs somehow from that which is computed by the formula by Masset and Bocquet-Appel (22.5 years). This is easily explainable as the life expectancy of the life table includes _all_ individuals. However, the value of 21.1 years is still in the range of 21.0 to 24.1 years (see indices) given by Masset and Bocquet-Appel.
 
+```{r}
+lt.representativity(lt.correction(schleswig)$life_table_corr)
+```
+
+Because the correction only applies to the youngest age group, it has no effects on the tests for representativity by Weiss and Masset and Bocquet-Appel. The correction, however, affects the test by Taylor et al. Now, the Total fertility rate based on the subadult index is much closer to the value computed from the individuals aged 15 and above. As the difference is below 0.692, the test now would not suspect a biased population. This might be taken as hint that the 'true' TFR-value is closer to 7 than to 4 and that the correction by Masset and Bocquet-Appel was both necessary and arrived at good estimates.
+
 ## References

vignettes/mortAAR_vignette_reproduction.Rmd

@@ -67,6 +67,8 @@ There are different approaches to calculate reproduction rates [e. g. @henneberg
 
 Recently, C. McFadden and M. F. Oxenham [-@mcfadden_oxenham_2018a] have published a formula to estimate the Total fertility rate from archaeological data, provided that infants are represented fully in the archaeological record. Unfortunately, this will not be the case for most archaeological datasets. Therefore, we used the data published by McFadden and Oxenham to apply it to the $P_{5-19}$-index after J.-P. Bocquet-Appel [-@bocquet_appel_2002]. We approximated the ratio by three different methods of fitting (linear, logistic, power) and recommend logistic fitting, but the others are available as well.
 
+Even more recently, B. Taylor, M. Oxenham and C. McFadden [-@taylor_et_al_2023] have presented a formula for the Total fertility rate for cases where pre-adult individuals are known to be underrepresented.
+
 The Gross reproduction rate (GRR) is calculated by multiplying the TFR with the ratio of female newborns, assumed to be a constant of 48.8% of all children [@hassan_1981, 136]. The Net reproduction rate is arrived at by summing the product of the GRR, the age specific fertility rate calculated from the data given by Hassan [-@hassan_1981, 137 tab. 8.7] and the age specific survival taken from the life table and dividing the result by 10000.
 
 The Rate of natural increase or Intrinsic growth rate `r` (growth in per cent per year) can be computed from the fertility calculations following Hassan [-@hassan_1981, 140]. An alternative way to calculate the intrinsic growth rate has recently described by C. McFadden and M. F. Oxenham [-@mcfadden_oxenham_2018b]. They present a regression calculation based on the index $D_{0-15}/D$ also used for fertility calculations (see above) in connections with modern data. It must not astonish that even with the McFadden/Oxenham-index used for the fertility rate, the actual numbers for the computed Intrinsic rate of growth and the Rate of Natural Increase can highly diverge, as with the former formula, further life table data is taken into account.
@@ -83,11 +85,17 @@ lt.reproduction(schleswig, fertility_rate = "McFO", growth_rate = "fertility")[c
 
 The McFadden/Oxenham-index gives a Total fertility rate of 4.85 which equals a total of around 5 children per woman. Of these children 48% will be female which leads to a Gross reproduction rate of 2.37. Taking into account the lived years of the Schleswig woman during their reproductive years, the Net reproduction rate would be 1.17, meaning that on average every woman -- regardless of their time of death -- would have had 1.17 daughters who in turn would have had children. This translates to a Rate of natural increase of 0.79%: Every year the Schleswig population would have increased by roughly 0.8% and with stable growth would have doubled every 88 years.
 
+```{r}
+lt.reproduction(schleswig, fertility_rate = "TOMc", growth_rate = "fertility")[c(-1,-2),]
+```
+
+The index of Taylor et al., which concentrates on the adult population only (age 15 and above), arrives at a much higher estimate for the Total fertility rate. This is computed with 6.92, and therefore also the Gross reproduction rate, the Net reproduction rate and the Rate of natural increase are higher than with the McFadden/Oxenham-index. Conversely, the doubling time of the population is with 27.04 years much lower.
+
 ```{r}
 lt.reproduction(lt.correction(schleswig)$life_table_corr, fertility_rate = "McFO", growth_rate = "fertility")[c(-1,-2),]
 ```
 
-As stated above, the index assumes, somehow unrealistically, that the youngest age-groups are fully represented in the skeletal population. If we apply the life table corrections as advised by C. Masset and J.-P. Bocquet-Appel via the function `lt.correction` (see vignette "Life table corrections"), the values change dramatically. The Total fertility and Gross reproduction rates increase, but because of the much higher mortality among the youngest, the Net reproduction rate and thus the Rate of natural increase shrink. The doubling time is extended to nearly 250 years.
+As stated above, the McFadden/Oxenham-index assumes, somehow unrealistically, that the youngest age-groups are fully represented in the skeletal population. If we apply the life table corrections as advised by C. Masset and J.-P. Bocquet-Appel via the function `lt.correction` (see vignette "Life table corrections"), the values change dramatically. The Total fertility and Gross reproduction rates increase, but because of the much higher mortality among the youngest, the Net reproduction rate and thus the Rate of natural increase shrink. The doubling time is extended to nearly 250 years.
 
 ```{r}
 lt.reproduction(schleswig, fertility_rate = "BA_log", growth_rate = "fertility")[c(-1,-2),]
@@ -113,7 +121,7 @@ lt.reproduction(schleswig, growth_rate = "McFO")[c(6,7),]
 
 The McFadden/Oxenham-index for the Rate of natural increase lies somewhere in-between. It gives a value of 1.81 and thus a doubling time of 38 years.
 
-In sum, the differing outcomes underline the importance of the fertility and the mortality among the youngest for the growth direction of a given population. Following Hassan [-@hassan_1981, 140], one would credit the corrected McFO-fertility rate to produce the most probable result as Hassan deems a growth rate of 0.5% per year as the maximum for prehistoric populations. On the other hand, also growth rates of 2.7% per year are known from ethnographic cases [@hassan_1981, 142].
+In sum, the differing outcomes underline the importance of the fertility and the mortality among the youngest for the growth direction of a given population. Following Hassan [-@hassan_1981, 140], one would credit the McFO-fertility index with lifetable correction after Masset and Bocquet-Appel to produce the most probable result as Hassan deems a growth rate of 0.5% per year as the maximum for prehistoric populations. On the other hand, also growth rates of 2.7% per year are known from ethnographic cases [@hassan_1981, 142].
 
 
 ### Rate of dependent individuals